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A006084 Continued fraction for e/3.

%I A006084 M4591 #27 May 17 2016 05:36:43
%S A006084 0,1,9,1,1,1,5,1,1,1,1,26,1,1,3,5,1,3,1,1,44,1,1,5,5,1,5,1,1,62,1,1,7,
%T A006084 5,1,7,1,1,80,1,1,9,5,1,9,1,1,98,1,1,11,5,1,11,1,1,116,1,1,13,5,1,13,
%U A006084 1,1,134,1,1,15,5,1,15,1,1,152,1,1,17,5,1,17,1,1,170,1,1,19,5,1,19,1,1
%N A006084 Continued fraction for e/3.
%D A006084 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 601.
%D A006084 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006084 Harry J. Smith, <a href="/A006084/b006084.txt">Table of n, a(n) for n = 1..20000</a>
%H A006084 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H A006084 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>
%H A006084 <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,-1).
%F A006084 First eight terms are 0, 1, 9, 1, 1, 1, 5, 1; then a(9k)=2k-1, a(9k+1)=1, a(9k+2)=1, a(9k+3)=18k+8, a(9k+4)=1, a(9k+5)=1, a(9k+6)=2k+1, a(9k+7)=5, a(9k+8)=1. - _Benoit Cloitre_, Apr 08 2003
%F A006084 G.f.: x^2*(1+9*x+x^2+x^3+x^4+5*x^5+x^6+x^7+x^8-x^9+8*x^10-x^11-x^12+x^13-5*x^14-x^15+x^16-x^17+x^19) / ((1-x)^2*(1+x+x^2)^2*(1+x^3+x^6)^2). - _Colin Barker_, May 16 2016
%e A006084 0.906093942819681745120095823... = 0 + 1/(1 + 1/(9 + 1/(1 + 1/(1 + ...)))). - _Harry J. Smith_, May 12 2009
%t A006084 ContinuedFraction[E/3,100] (* _Harvey P. Dale_, Oct 14 2013 *)
%t A006084 Join[{0, 1, 9},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -1},{1, 1, 1, 5, 1, 1, 1, 1, 26, 1, 1, 3, 5, 1, 3, 1, 1, 44},89]] (* _Ray Chandler_, Sep 03 2015 *)
%o A006084 (PARI) { allocatemem(932245000); default(realprecision, 54000); x=contfrac(exp(1)/3); for (n=1, 20000, write("b006084.txt", n, " ", x[n])); } \\ _Harry J. Smith_, May 12 2009
%o A006084 (PARI) concat(0, Vec(x^2*(1+9*x+x^2+x^3+x^4+5*x^5+x^6+x^7+x^8-x^9+8*x^10-x^11-x^12+x^13-5*x^14-x^15+x^16-x^17+x^19)/((1-x)^2*(1+x+x^2)^2*(1+x^3+x^6)^2) + O(x^50))) \\ _Colin Barker_, May 16 2016
%Y A006084 Cf. A019740 = Decimal expansion. - _Harry J. Smith_, May 12 2009
%K A006084 nonn,cofr,easy
%O A006084 1,3
%A A006084 _N. J. A. Sloane_, _Jeffrey Shallit_
%E A006084 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003